Can a single F-curve be bound to several properties?
I would do this with drivers. I don't know if there's a way to make an infinite driver value range, but in this case, I just set it up to be 1000, which is far too big for blender to display the default cube. 

Select the object in question (in this case, the default cube) and go to the object panel. Towards the bottom, you will see a section called "custom properties." Expand that and create a new property. I named mine "scales" to separate it from pre-made properties. You can set the upper and lower limit to however much you want. I used 0.0 and 1000.0 in mine. Make sure you make it a float, or else you will only be able to use whole numbers (a float is a decimal point, ie 1.0, 1.5, 1.0004, etc).

Create a driver on one of the scale axes (you only need one, which you can copy/paste to the rest). Right click on the channel value slider and create single driver. 

Split off the viewport into a graph editor, then switch from F-curve display to Drivers display (the red box on the lower left). This will allow you to edit the driver and its data. Open the properties region, just as you would in the viewport. There, you'll see the drivers in their data form. 

Now that the UI is set up, go back to the custom properties of your cube, right click on the value slider bar of the property you set up and copy data path (roughly half-way down the list). 

In the driver setup in the graph, change the "type" to one of the options with "value" in it (ie minimum value, maximum value, sum values, average value). In the variable (the lower half of that driver setup area), change the default "transform channel" to "single property." Change the property type to the object (in my case, the default cube is an object, so i selected object), then find your object in the menu beside it (mine is "cube"). 

Remember that data you copied from your custom properties? Paste that into the "Path" field below. What it should change to is a small DNA-like symbol, which indicates your link is active. To be safe, click "update dependencies" at the top of the driver data area. Copy that driver to the rest of the channels and you'll be ready to go.

Slide the custom property around to test it. To animate the property, just key and edit the custom property just as you would a transform channel on a bone. It appears in the graph editor and since the driver interpolation is linear, eases are intuitive and very simple.